In a medium bowl, soak green onions in ice water to keep crisp. In a large heavy-bottomed saucepan, pour enough oil to fill the pan about a third of the way. Heat over medium heat until a deep-frying thermometer inserted in the oil reaches 400 degrees F. Add half of the julienned taro root to the oil. Fry about 2 minutes, or until crisp and light gold in color. Remove taro slices from oil, drain on paper towels, and lightly salt. Repeat with remaining taro root.
Bring oil back up to 400 degrees F. Add half the vermicelli to the oil and fry until the noodles start to curl, about 2 or 3 seconds.. Remove the vermicelli from the oil, drain on paper towel and lightly salt. Fry the remaining vermicelli.
Remove the julienned green onion from the ice water and drain on paper towels. Divide the vermicelli equally among 4 serving plates. Divide and arrange the julienned green onions, cucumber, carrot, jicama, and diced tomatoes evenly over vermicelli. Top salads with the pickled onions and the fried taro root.
Sprinkle 1 teaspoon toasted sesame seeds and 1 1/2 teaspoons crushed peanuts over each salad. In a small bowl, combine the edible flower petals, fennel seedlings, daikon sprouts, and fried shallots. Divide the flower-sprout-shallot mixture evenly among salad plates. Serve the salads with the Salted Apricot Dressing alongside.
Pickled Red Onion
In a small saucepan add the rice wine vinegar and water and over medium heat, bring the mixture to a boil. Add the salt, black peppercorns, fennel seeds, bay leaf, and thyme to the saucepan and boil another 5 minutes. In a medium nonreactive bowl, combine the julienned red onion and the hot vinegar mixture. Toss gently to coat and let sit for at least 1 hour.
Salted Apricot Dressing
In a blender, add the salted apricot paste, rice wine vinegar, mirin, Dashi, Onion Oil, sugar, ginger, and the salt. Blend until smooth, about 30 seconds.